The couple are “desperate” to have children “as soon as possible”, a source close to the couple said.
It comes after William and the Duchess of Cambridge’s third child, Louis Arthur Charles of Cambridge, was born at the end of April.
Speaking ahead of Harry and Meghan’s wedding later this month, a friend said: “They are desperate for children as soon as possible, they hope it won’t take long.”
During an interview after his engagement to former Suits star Ms Markle, Harry said: “hopefully we’ll start a family in the near future.”
Harry and former actress Ms Markle’s wedding is on Saturday, May 19 at St George’s Chapel at Windsor Castle at 12pm.
Prince George and Princess Charlotte are expected to be involved as page boy and flower girl.
The wedding is expected to be much smaller than the Duke and Duchess of Cambridge’s ceremony, which saw 1,900 arrive at the church.
Some 600 guests are expected to attend.